The nesting of two or more functions to form a single new function is known as composition. The composition
of two functions
and is denoted , where is a function whose domain includes the range of . The notation
(1)
is sometimes used to explicitly indicate the variable.
Composition is associative, so that
(2)
If the functions
is continuous at
and is continuous at , then is also continuous at .
A function
which is the composition of two other functions, say and , is sometimes said to be a composite function.
A combinatorial composition is defined as an ordered arrangement of nonnegative integers which sum to (Skiena 1990, p. 60). It is therefore a partition
in which order is significant. For example, there are eight compositions of 4,
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
A positive integer
has compositions.
The number of compositions of into parts (where 0 is not allowed as a part) is given by
(11)
(12)
The number
of compositions of a number of length (where 0 is allowed) is given by the formula
(13)
(14)
which is implemented as NumberOfCompositions[n, k] in the Wolfram Language
package Combinatorica` . The following table gives counts of compositions
where 0 is allowable.
An operation called composition is also defined on binary quadratic forms. For two numbers represented by two forms, the product can then
be represented by the composition. For example, the composition of the forms and is given by , and in this case, the product of 17 and 13 would
be represented as (). There are several algorithms for computing
binary quadratic form composition, which is the basis for some factoring methods.